Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C73 · Schur multiplier (H2) = C23 |
| direct product | C73×#9530 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | #386 |
| commutator | C22×C4 | C22×C4×C73 |
| Frattini, cosocle | C2×C42 | C23×C73 |
| prime radical | C22×D4×C73 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C2×C42×C73 ◁ G | |
428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 105 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 3 generators on 105 points:
(2,6)(3,10)(4,7)(9,14)(11,12)(13,15)(17,18,21,25,22,19,23,27)(20,26,29,32,28,24,30,31)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105);(1,2,7,14,5,9,4,6)(3,8,10,15,12,16,11,13)(17,19,21,27,22,18,23,25)(20,26,30,31,28,24,29,32);(1,3,5,12)(2,8,9,16)(4,11,7,10)(6,13,14,15)(17,20)(18,24)(19,26)(21,30)(22,28)(23,29)(25,32)(27,31)2701 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 2518);